Orville is checking whether the two functions $f(x) = \sqrt[3]{4x - 2}$ and $g(x) = (x + 24)^3$ are inverses of each other. To verify this, we would typically compute both $f(g(x))$ and $g(f(x))$ and see if they both return $x$, indicating that the functions undo each other.

Let’s examine his steps:

Step 1: Checking $f \circ g$ (i.e., $f(g(x))$)

1. Starting with $g(x) = (x + 24)^3$: He correctly substitutes $g(x)$ into $f(x)$ for the composition: $f(g(x)) = f((x + 24)^3)$

2. Substitute into $f(x)$: Since $f(x) = \sqrt[3]{4x - 2}$, we substitute $(x + 24)^3$ into this formula: $f((x + 24)^3) = \sqrt[3]{4((x + 24)^3) - 2}$

   However, Orville makes an error here by simplifying incorrectly. The correct expression should remain as: $f(g(x)) = \sqrt[3]{4(x + 24)^3 - 2}$

   This cannot be simplified further without additional steps, but Orville simplifies it inappropriately in his next steps, leading to an incorrect conclusion.

Conclusion:

Orville's simplification contains errors. He should not have simplified $f(g(x))$ as $x$, since the functions are not inverses in the form they are given. Therefore, the correct answer is:

Orville made an error in simplifying $f(g(x))$. The two functions are not inverses of each other.

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Do you want more details or have any questions?

Here are five questions you might explore next:

1. How do you properly check if two functions are inverses of each other?
2. What are the algebraic steps needed to simplify $f(g(x))$ correctly?
3. Can you find $g(f(x))$ and verify if it returns $x$?
4. What properties must a function have to be invertible?
5. How can you graphically verify whether two functions are inverses?

Tip: When simplifying compositions of functions, make sure to apply operations step by step without assuming results early in the process.